The call to avr/io.h was shown in the compiler error statement as coming from line 25 in twi.c.
The compiler is looking for the old version.C:\Users\xxxx\Documents\Arduino\libraries\Wire\utility\twi.c:25: fatal error: avr/io.h: No such file or directory
My scheme shown above did not work entirely. It did prove that the Due board is named "__SAM3X8E__" as it looked for sam/pio.h. However, it did not find that file either so the solution will not work.